Abstract
We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime $2$, hence for characters of odd degree.
-
[Bo05]
C. Bonnafé, "Quasi-isolated elements in reductive groups," Comm. Algebra, vol. 33, iss. 7, pp. 2315-2337, 2005.
@ARTICLE{Bo05, mrkey = {2153225},
number = {7},
issn = {0092-7872},
author = {Bonnaf{é},
C{é}dric},
mrclass = {20G15},
doi = {10.1081/AGB-200063602},
journal = {Comm. Algebra},
zblnumber = {1096.20037},
volume = {33},
mrnumber = {2153225},
fjournal = {Communications in Algebra},
mrreviewer = {Evgenii L. Bashkirov},
coden = {COALDM},
title = {Quasi-isolated elements in reductive groups},
year = {2005},
pages = {2315--2337},
} -
[Cedric] C. Bonnafé, Sur les Caractères des Groupes Réductifs finis à Centre Non Connexe: Applications aux Groupes Spéciaux Linéaires et Unitaires, Paris: Math. Soc. France, 2006, vol. 306.
@BOOK{Cedric, mrkey = {2274998},
issn = {0303-1179},
author = {Bonnaf{é},
C{é}dric},
mrclass = {20G40 (20G05)},
series = {Astérisque},
address = {Paris},
publisher = {Math. Soc. France},
zblnumber = {1157.20022},
volume = {306},
mrnumber = {2274998},
fjournal = {Astérisque},
mrreviewer = {Robert B{é}dard},
title = {Sur les Caractères des Groupes Réductifs finis à Centre Non Connexe: Applications aux Groupes Spéciaux Linéaires et Unitaires},
year = {2006},
pages = {vi+165},
} -
[BM92]
M. Broué and G. Malle, "Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis," Math. Ann., vol. 292, iss. 2, pp. 241-262, 1992.
@ARTICLE{BM92, mrkey = {1149033},
number = {2},
issn = {0025-5831},
author = {Brou{é},
Michel and Malle, Gunter},
mrclass = {20G40},
doi = {10.1007/BF01444619},
journal = {Math. Ann.},
zblnumber = {0820.20057},
volume = {292},
mrnumber = {1149033},
fjournal = {Mathematische Annalen},
mrreviewer = {Bhama Srinivasan},
coden = {MAANA},
title = {Théorèmes de {S}ylow génériques pour les groupes réductifs sur les corps finis},
year = {1992},
pages = {241--262},
} -
[BMM] M. Broué, G. Malle, and J. Michel, "Generic blocks of finite reductive groups," in Représentations Unipotentes Génériques et Blocs des Groupes Réductifs Finis, Paris: Math. Soc. France, 1993, vol. 212, pp. 7-92.
@INCOLLECTION{BMM, mrkey = {1235832},
issn = {0303-1179},
author = {Brou{é},
Michel and Malle, Gunter and Michel, Jean},
mrclass = {20G05 (20C20)},
series = {Astérisque},
address = {Paris},
publisher = {Math. Soc. France},
zblnumber = {0843.20012},
volume = {212},
mrnumber = {1235832},
booktitle = {Repr{é}sentations Unipotentes G{é}n{é}riques et Blocs des Groupes R{é}ductifs Finis},
fjournal = {Astérisque},
mrreviewer = {Bhama Srinivasan},
title = {Generic blocks of finite reductive groups},
year = {1993},
pages = {7--92},
} -
[CE04]
M. Cabanes and M. Enguehard, Representation Theory of Finite Reductive Groups, Cambridge: Cambridge Univ. Press, 2004, vol. 1.
@BOOK{CE04, mrkey = {2057756},
author = {Cabanes, Marc and Enguehard, Michel},
mrclass = {20G05 (20G40)},
series = {New Mathematical Monographs},
isbn = {0-521-82517-2},
address = {Cambridge},
publisher = {Cambridge Univ. Press},
doi = {10.1017/CBO9780511542763},
zblnumber = {1069.20032},
volume = {1},
mrnumber = {2057756},
mrreviewer = {Bhama Srinivasan},
title = {Representation Theory of Finite Reductive Groups},
year = {2004},
pages = {xviii+436},
} -
[CS13]
M. Cabanes and B. Späth, "Equivariance and extendibility in finite reductive groups with connected center," Math. Z., vol. 275, iss. 3-4, pp. 689-713, 2013.
@ARTICLE{CS13, mrkey = {3127033},
number = {3-4},
issn = {0025-5874},
author = {Cabanes, Marc and Sp{ä}th, Britta},
mrclass = {20C33 (20C15 20C20 20D06 20G40)},
doi = {10.1007/s00209-013-1156-7},
journal = {Math. Z.},
zblnumber = {1328.20019},
volume = {275},
mrnumber = {3127033},
fjournal = {Mathematische Zeitschrift},
mrreviewer = {George J. McNinch},
title = {Equivariance and extendibility in finite reductive groups with connected center},
year = {2013},
pages = {689--713},
} -
[CS15] M. Cabanes and B. Späth, "Equivariant character correspondences and inductive McKay condition for type A," J. reine angew. Math., 2016.
@article{CS15,
author = {Cabanes, Marc and Sp{ä}th, Britta},
TITLE = {Equivariant character correspondences and inductive {M}c{K}ay condition for type {A}},
journal = {J. reine angew. Math.},
note={to appear},
year = {2016},
} -
[Ca85] R. W. Carter, Finite Groups of Lie Type. Conjugacy classes and complex characters, New York: John Wiley & Sons, 1985.
@BOOK{Ca85, mrkey = {0794307},
author = {Carter, Roger W.},
mrclass = {20G40 (20-02 20C15)},
series = {Pure Appl. Math.},
isbn = {0-471-90554-2},
address = {New York},
publisher = {John Wiley \& Sons},
mrnumber = {0794307},
mrreviewer = {David B. Surowski},
zblnumber = {0567.20023},
title = {Finite Groups of {L}ie Type. Conjugacy classes and complex characters},
year = {1985},
pages = {xii+544},
} -
[DM]
F. Digne and J. Michel, Representations of Finite Groups of Lie Type, Cambridge: Cambridge Univ. Press, 1991, vol. 21.
@BOOK{DM, mrkey = {1118841},
author = {Digne, Fran{ç}ois and Michel, Jean},
mrclass = {20G05 (20-02 20C33 20G40)},
series = {London Math. Soc. Stud. Texts},
isbn = {0-521-40117-8; 0-521-40648-X},
address = {Cambridge},
publisher = {Cambridge Univ. Press},
doi = {10.1017/CBO9781139172417},
zblnumber = {0815.20014},
volume = {21},
mrnumber = {1118841},
mrreviewer = {Bhama Srinivasan},
title = {Representations of Finite Groups of {L}ie Type},
year = {1991},
pages = {iv+159},
} -
[DD93]
R. Dipper and J. Du, "Harish-Chandra vertices," J. Reine Angew. Math., vol. 437, pp. 101-130, 1993.
@ARTICLE{DD93, mrkey = {1212254},
issn = {0075-4102},
author = {Dipper, Richard and Du, Jie},
mrclass = {20C20},
doi = {10.1515/crll.1993.437.101},
journal = {J. Reine Angew. Math.},
zblnumber = {0768.20004},
volume = {437},
mrnumber = {1212254},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
mrreviewer = {Bhama Srinivasan},
title = {Harish-{C}handra vertices},
year = {1993},
pages = {101--130},
} -
[GeckHC]
M. Geck, "A note on Harish-Chandra induction," Manuscripta Math., vol. 80, iss. 4, pp. 393-401, 1993.
@ARTICLE{GeckHC, mrkey = {1243154},
number = {4},
issn = {0025-2611},
author = {Geck, Meinolf},
mrclass = {20C33 (20G05)},
doi = {10.1007/BF03026560},
journal = {Manuscripta Math.},
zblnumber = {0821.20027},
volume = {80},
mrnumber = {1243154},
fjournal = {Manuscripta Mathematica},
mrreviewer = {Robert B. Howlett},
coden = {MSMHB2},
title = {A note on {H}arish-{C}handra induction},
year = {1993},
pages = {393--401},
} -
[GP] M. Geck and G. Pfeiffer, Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, New York: The Clarendon Press, Oxford University Press, 2000.
@BOOK{GP, mrkey = {1778802},
author = {Geck, Meinolf and Pfeiffer, G{ö}tz},
mrclass = {20C15 (20C08 20F55)},
series = {London Math. Soc. Monogr.},
isbn = {0-19-850250-8},
publisher = {The Clarendon Press, Oxford University Press},
zblnumber = {0996.20004},
address = {New York},
mrnumber = {1778802},
mrreviewer = {Jian-yi Shi},
title = {Characters of Finite {C}oxeter {G}roups and {I}wahori-{H}ecke Algebras},
year = {2000},
pages = {xvi+446},
} -
[GLS3]
D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Providence, RI: Amer. Math. Soc., 1994, vol. 40.
@BOOK{GLS3, mrkey = {1303592},
author = {Gorenstein, Daniel and Lyons, Richard and Solomon, Ronald},
mrclass = {20D05 (20-02)},
series = {Math. Surveys Monogr.},
isbn = {0-8218-0334-4},
address = {Providence, RI},
publisher = {Amer. Math. Soc.},
doi = {10.1090/surv/040.1},
zblnumber = {0816.20016},
volume = {40},
mrnumber = {1303592},
mrreviewer = {J. L. Alperin},
title = {The Classification of the Finite Simple Groups},
year = {1994},
pages = {xiv+165},
} -
[HL]
R. B. Howlett and G. I. Lehrer, "Induced cuspidal representations and generalised Hecke rings," Invent. Math., vol. 58, iss. 1, pp. 37-64, 1980.
@ARTICLE{HL, mrkey = {0570873},
number = {1},
issn = {0020-9910},
author = {Howlett, R. B. and Lehrer, G. I.},
mrclass = {20C15 (20G40)},
doi = {10.1007/BF01402273},
journal = {Invent. Math.},
zblnumber = {0435.20023},
volume = {58},
mrnumber = {0570873},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Bhama Srinivasan},
coden = {INVMBH},
title = {Induced cuspidal representations and generalised {H}ecke rings},
year = {1980},
pages = {37--64},
} -
[HL83]
R. B. Howlett and G. I. Lehrer, "Representations of generic algebras and finite groups of Lie type," Trans. Amer. Math. Soc., vol. 280, iss. 2, pp. 753-779, 1983.
@ARTICLE{HL83, mrkey = {0716849},
number = {2},
issn = {0002-9947},
author = {Howlett, R. B. and Lehrer, G. I.},
mrclass = {20G05 (16A64 16A65)},
doi = {10.2307/1999645},
journal = {Trans. Amer. Math. Soc.},
zblnumber = {0537.20018},
volume = {280},
mrnumber = {0716849},
fjournal = {Transactions of the American Mathematical Society},
mrreviewer = {S. Yu. Prishepionok},
coden = {TAMTAM},
title = {Representations of generic algebras and finite groups of {L}ie type},
year = {1983},
pages = {753--779},
} -
[HL94]
R. B. Howlett and G. I. Lehrer, "On Harish-Chandra induction and restriction for modules of Levi subgroups," J. Algebra, vol. 165, iss. 1, pp. 172-183, 1994.
@ARTICLE{HL94, mrkey = {1272585},
number = {1},
issn = {0021-8693},
author = {Howlett, R. B. and Lehrer, G. I.},
mrclass = {20C33 (20C20)},
doi = {10.1006/jabr.1994.1104},
journal = {J. Algebra},
zblnumber = {0802.20036},
volume = {165},
mrnumber = {1272585},
fjournal = {Journal of Algebra},
mrreviewer = {Peter Fleischmann},
coden = {JALGA4},
title = {On {H}arish-{C}handra induction and restriction for modules of {L}evi subgroups},
year = {1994},
pages = {172--183},
} -
@BOOK{Hu, mrkey = {1645304},
number = {25},
author = {Huppert, Bertram},
mrclass = {20C15 (20C05 20C10)},
series = {de Gruyter Exp. Math.},
isbn = {3-11-015421-8},
address = {Berlin},
publisher = {Walter de Gruyter \& Co.},
doi = {10.1515/9783110809237},
zblnumber = {0932.20007},
mrnumber = {1645304},
mrreviewer = {Gerhard Hiss},
title = {Character Theory of Finite Groups},
year = {1998},
pages = {vi+618},
} -
[Isa] M. I. Isaacs, Character Theory of Finite Groups, New York: Academic Press [Harcourt Brace Jovanovich, Publishers], 1976, vol. 69.
@BOOK{Isa, mrkey = {0460423},
author = {Isaacs, I. Martin},
mrclass = {20-02},
series = {Pure Appl. Math.},
address = {New York},
publisher = {Academic Press [Harcourt Brace Jovanovich, Publishers]},
volume = {69},
mrnumber = {0460423},
mrreviewer = {Stephen D. Smith},
zblnumber = {0337.20005},
title = {Character Theory of Finite Groups},
year = {1976},
pages = {xii+303},
} -
[IMN]
M. I. Isaacs, G. Malle, and G. Navarro, "A reduction theorem for the McKay conjecture," Invent. Math., vol. 170, iss. 1, pp. 33-101, 2007.
@ARTICLE{IMN, mrkey = {2336079},
number = {1},
issn = {0020-9910},
author = {Isaacs, I. Martin and Malle, Gunter and Navarro, Gabriel},
mrclass = {20C20 (20C33 20C34)},
doi = {10.1007/s00222-007-0057-y},
journal = {Invent. Math.},
zblnumber = {1138.20010},
volume = {170},
mrnumber = {2336079},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Charles W. Eaton},
coden = {INVMBH},
title = {A reduction theorem for the {M}c{K}ay conjecture},
year = {2007},
pages = {33--101},
} -
[Lu] G. Lusztig, Characters of Reductive Groups over a Finite Field, Princeton, NJ: Princeton Univ. Press, 1984, vol. 107.
@BOOK{Lu, mrkey = {0742472},
author = {Lusztig, George},
mrclass = {20G05 (14L20 20C15)},
series = {Ann. of Math. Stud.},
isbn = {0-691-08350-9; 0-691-08351-7},
address = {Princeton, NJ},
publisher = {Princeton Univ. Press},
zblnumber = {0556.20033},
volume = {107},
mrnumber = {0742472},
mrreviewer = {Bhama Srinivasan},
title = {Characters of Reductive Groups over a Finite Field},
year = {1984},
pages = {xxi+384},
} -
[LuDis] G. Lusztig, "On the representations of reductive groups with disconnected centre," in Orbites Unipotentes et Représentations, I, Paris: Math. Soc. France, 1988, vol. 168, p. 10, 157-166.
@incollection{LuDis, mrkey = {1021495},
issn = {0303-1179},
author = {Lusztig, George},
mrclass = {20G05 (20G40)},
series = {Astérisque},
address = {Paris},
publisher = {Math. Soc. France},
volume = {168},
mrnumber = {1021495},
booktitle = {Orbites Unipotentes et Repr{é}sentations, I},
mrreviewer = {Robert B{é}dard},
title = {On the representations of reductive groups with disconnected centre},
year = {1988},
pages = {10, 157--166},
} -
[MaH0]
G. Malle, "Height 0 characters of finite groups of Lie type," Represent. Theory, vol. 11, pp. 192-220, 2007.
@ARTICLE{MaH0, mrkey = {2365640},
issn = {1088-4165},
author = {Malle, Gunter},
mrclass = {20C33 (20G40)},
doi = {10.1090/S1088-4165-07-00312-3},
journal = {Represent. Theory},
zblnumber = {1139.20008},
volume = {11},
mrnumber = {2365640},
fjournal = {Representation Theory. An Electronic Journal of the American Mathematical Society},
mrreviewer = {Gerhard Hiss},
title = {Height 0 characters of finite groups of {L}ie type},
year = {2007},
pages = {192--220},
} -
[ManonLie]
G. Malle, "The inductive McKay condition for simple groups not of Lie type," Comm. Algebra, vol. 36, iss. 2, pp. 455-463, 2008.
@ARTICLE{ManonLie, mrkey = {2387535},
number = {2},
issn = {0092-7872},
author = {Malle, Gunter},
mrclass = {20C15 (20C33 20C34)},
doi = {10.1080/00927870701716090},
journal = {Comm. Algebra},
zblnumber = {1153.20009},
volume = {36},
mrnumber = {2387535},
fjournal = {Communications in Algebra},
mrreviewer = {A. S. Kondrat{\cprime}ev},
coden = {COALDM},
title = {The inductive {M}c{K}ay condition for simple groups not of {L}ie type},
year = {2008},
pages = {455--463},
} -
[MaExt]
G. Malle, "Extensions of unipotent characters and the inductive McKay condition," J. Algebra, vol. 320, iss. 7, pp. 2963-2980, 2008.
@ARTICLE{MaExt, mrkey = {2442005},
number = {7},
issn = {0021-8693},
author = {Malle, Gunter},
mrclass = {20C33 (20C15)},
doi = {10.1016/j.jalgebra.2008.06.033},
journal = {J. Algebra},
zblnumber = {1163.20003},
volume = {320},
mrnumber = {2442005},
fjournal = {Journal of Algebra},
mrreviewer = {A. S. Kondrat{\cprime}ev},
coden = {JALGA4},
title = {Extensions of unipotent characters and the inductive {M}c{K}ay condition},
year = {2008},
pages = {2963--2980},
} -
[MT]
G. Malle and D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge: Cambridge Univ. Press, 2011, vol. 133.
@BOOK{MT, mrkey = {2850737},
author = {Malle, Gunter and Testerman, Donna},
mrclass = {20G15 (20D06 20G05)},
series = {Cambridge Studies in Advanced Mathematics},
isbn = {978-1-107-00854-0},
address = {Cambridge},
publisher = {Cambridge Univ. Press},
doi = {10.1017/CBO9780511994777},
zblnumber = {1256.20045},
volume = {133},
mrnumber = {2850737},
mrreviewer = {Timothy C. Burness},
title = {Linear Algebraic Groups and Finite Groups of {L}ie Type},
year = {2011},
pages = {xiv+309},
} -
[McGovern]
K. McGovern, "Multiplicities of principal series representations of finite groups with split $(B,\,N)$-pairs," J. Algebra, vol. 77, iss. 2, pp. 419-442, 1982.
@ARTICLE{McGovern, mrkey = {0673126},
number = {2},
issn = {0021-8693},
author = {McGovern, Kathy},
mrclass = {20G05},
doi = {10.1016/0021-8693(82)90264-2},
journal = {J. Algebra},
zblnumber = {0492.20023},
volume = {77},
mrnumber = {0673126},
fjournal = {Journal of Algebra},
mrreviewer = {Louis Solomon},
title = {Multiplicities of principal series representations of finite groups with split {$(B,\,N)$}-pairs},
year = {1982},
pages = {419--442},
} -
[McK]
J. McKay, "Irreducible representations of odd degree," J. Algebra, vol. 20, pp. 416-418, 1972.
@ARTICLE{McK, mrkey = {0286904},
issn = {0021-8693},
author = {McKay, John},
mrclass = {20.80},
doi = {10.1016/0021-8693(72)90066-X},
journal = {J. Algebra},
zblnumber = {0235.20009},
volume = {20},
mrnumber = {0286904},
fjournal = {Journal of Algebra},
mrreviewer = {J. S. Frame},
title = {Irreducible representations of odd degree},
year = {1972},
pages = {416--418},
} -
[NT15]
G. Navarro and P. H. Tiep, "Irreducible representations of odd degree," Math. Ann., vol. 365, pp. 1155-1185, 2016.
@ARTICLE{NT15,
author = {Navarro, G. and Tiep, P. H.},
journal = {Math.~Ann.},
volume = {365},
pages = {1155--1185},
title = {Irreducible representations of odd degree},
year = {2016},
doi = {10.1007/s00208-015-1334-5},
MRNUMBER={3521086},
} -
@MISC{Sp_Diss,
author = {Sp{ä}th, Britta},
note = {dissertation, {TU K}aiserslautern},
url = {http://kluedo.ub.uni-kl.de/volltexte/2007/2073/pdf/Spaeth\_McKay.pdf},
title = {Die {M}cKay-{V}ermutung für quasi-einfache {G}ruppen vom {L}ie-{T}yp},
} -
[Sp09]
B. Späth, "The McKay conjecture for exceptional groups and odd primes," Math. Z., vol. 261, iss. 3, pp. 571-595, 2009.
@ARTICLE{Sp09, mrkey = {2471089},
number = {3},
issn = {0025-5874},
author = {Sp{ä}th, Britta},
mrclass = {20C20 (20G40)},
doi = {10.1007/s00209-008-0340-7},
journal = {Math. Z.},
zblnumber = {1178.20006},
volume = {261},
mrnumber = {2471089},
fjournal = {Mathematische Zeitschrift},
mrreviewer = {Stuart Martin},
coden = {MAZEAX},
title = {The {M}c{K}ay conjecture for exceptional groups and odd primes},
year = {2009},
pages = {571--595},
} -
[Sp13]
B. Späth, "A reduction theorem for the Alperin-McKay conjecture," J. Reine Angew. Math., vol. 680, pp. 153-189, 2013.
@ARTICLE{Sp13, mrkey = {3100954},
issn = {0075-4102},
author = {Sp{ä}th, Britta},
mrclass = {20C20 (20C15 20C33)},
doi = {10.1515/crelle.2012.035},
journal = {J. Reine Angew. Math.},
zblnumber = {1283.20006},
volume = {680},
mrnumber = {3100954},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
mrreviewer = {Burkhard K{ü}lshammer},
title = {A reduction theorem for the {A}lperin-{M}c{K}ay conjecture},
year = {2013},
pages = {153--189},
} -
[Spaeth2]
B. Späth, "Sylow $d$-tori of classical groups and the McKay conjecture. I," J. Algebra, vol. 323, iss. 9, pp. 2469-2493, 2010.
@ARTICLE{Spaeth2, mrkey = {2602390},
number = {9},
issn = {0021-8693},
author = {Sp{ä}th, Britta},
mrclass = {20C33 (20C20 20G05 20G40)},
doi = {10.1016/j.jalgebra.2010.02.008},
journal = {J. Algebra},
zblnumber = {1260.20015},
volume = {323},
mrnumber = {2602390},
fjournal = {Journal of Algebra},
mrreviewer = {George J. McNinch},
coden = {JALGA4},
title = {Sylow {$d$}-tori of classical groups and the {M}c{K}ay conjecture. {I}},
year = {2010},
pages = {2469--2493},
} -
[Sp12]
B. Späth, "Inductive McKay condition in defining characteristic," Bull. Lond. Math. Soc., vol. 44, iss. 3, pp. 426-438, 2012.
@ARTICLE{Sp12, mrkey = {2966987},
number = {3},
issn = {0024-6093},
author = {Sp{ä}th, Britta},
mrclass = {20C33 (20C15)},
doi = {10.1112/blms/bdr100},
journal = {Bull. Lond. Math. Soc.},
zblnumber = {1251.20020},
volume = {44},
mrnumber = {2966987},
fjournal = {Bulletin of the London Mathematical Society},
mrreviewer = {Hung Phi Tong-Viet},
title = {Inductive {M}c{K}ay condition in defining characteristic},
year = {2012},
pages = {426--438},
} -
[Springer]
T. A. Springer, "Regular elements of finite reflection groups," Invent. Math., vol. 25, pp. 159-198, 1974.
@ARTICLE{Springer, mrkey = {0354894},
issn = {0020-9910},
author = {Springer, T. A.},
mrclass = {20H15},
doi = {10.1007/BF01390173},
journal = {Invent. Math.},
zblnumber = {0287.20043},
volume = {25},
mrnumber = {0354894},
fjournal = {Inventiones Mathematicae},
mrreviewer = {R. Steinberg},
title = {Regular elements of finite reflection groups},
year = {1974},
pages = {159--198},
} -
[Tits]
J. Tits, "Normalisateurs de tores. I. Groupes de Coxeter étendus," J. Algebra, vol. 4, pp. 96-116, 1966.
@ARTICLE{Tits, mrkey = {0206117},
issn = {0021-8693},
author = {Tits, J.},
mrclass = {20.75},
doi = {10.1016/0021-8693(66)90053-6},
journal = {J. Algebra},
zblnumber = {0145.24703},
volume = {4},
mrnumber = {0206117},
fjournal = {Journal of Algebra},
mrreviewer = {Rimhak Ree},
title = {Normalisateurs de tores. {I}. {G}roupes de {C}oxeter étendus},
year = {1966},
pages = {96--116},
}